Hi Carl,
You just pointed out one of the reasons I don't always like having a .co.uk and a .com. It really does depend on each business. I'll do the pros and cons for you and hope that helps. Both ways will work, it's going to come down to your preferences. And don't go by how many there are, just look for what matters to you.
Pros of One Domain:
- You can geo-target subdirectories (domain.com/uk) and have specific product pages for that content that needs to be different but share the overlapping content like About, Blog, and other material that doesn't change.
- Domain authority condenses and gets stronger overall over time. All of your marketing is to one site rather than two. This is paired with the Pro before are why I'd go down this route if I were you.
- Don't have to rewrite all content like the blogs and articles if they don't need it.
- Expansions to other countries later would rank faster since they would be a part of the main domain rather than a brand new one (ccTLD) every time.
Cons of One Domain:
- Regional differences (personalized vs personalised) cannot be accounted for in all site content. This might not be a problem for your audience though. You'll still need to maintain and write region specific content sometimes, but not as much as the other way.
- Users outside the US or UK could land on either product versions depending on a number of issues. Again, this might not matter to you. You might not want that traffic.
- It'll take some time to get the targeting right, be prepared for the change over and seeing US/UK content ranking over the other sometimes. It happens, but the search engines get it right about 80-90% of the time in the wild.
Pros of Two Domains
- Can target content, all content on the users in that area. Only useful if that is needed.
- ccTLDs are the strongest country specific marker. It is not make or break though.
Cons of Two Domains
- Maintaining two sites, two domain names, two sets of content. Cost and time factors.
- Marketing needs to be done separately for each site.

This is highly confounding. I've crawled as Bingbot and nothing is in it's way.